概率论与数理统计(英文) 第六章.doc

（-24 | 25-）6. Fundamental Sampling Distributions and Data Descriptions 抽样分布6.1 Analysis of Data Mean 均值 , Median 中位数 Mean Deviation 平均差（均值离差） 方差 标准差 62 Random Sampling Sampling is one of the most important concepts in the study of statistics. We need the fundamental ideas of populations and samples before studying particular statistical descriptions. population (总体) sample （样本） sampling （抽样）Definition 6.2.1 A population is the set of data or measurements consists of all conceivably possible observations from all objects in a given phenomenon. .A population may consist of finitely or infinitely many varieties. For example, in the study of the grade of Calculus course in some university of the year 2006, the grades of all the students who took this course constitute the population. a finite population（有限总体）infinite population（无限总体）This is a finite population. A example is the study of the length of newborns in China. The population is then the all possible lengths of the newborns in China, in the past, now or in the future. Such population is an infinite one. Since in many cases we are not able to investigate a whole population we are obliged to get conclusions regarding a population from its samples.sample （样本、子样）Definition 6.2.2 A sample is a subset of the population from which people can draw conclusions about the whole. sampling(抽样)taking a sample: The process of performing an experiment to obtain a sample from the population is called sampling. sampling is done with replacement 有放回抽样 sampling is done without replacement 无放回 (有放回抽样，使样本点独立同分布)The purpose of the sampling is to find out something about the nature of the population.good samplegood estimatesgood estimates concerning a population necessitate good sample. good sample-random sample (随机样本)Definition 6.2.3 A random sample (some times referred to also as a simple random sample) of size of a random variable is a collection of random variables, which are independent and each of them is identically distributed with the population random variable . If is the value of the joint distribution of such a set of random variables at , we can get that (6.1.1)in which is the population distribution at . Definition 6.1.4 A random sample of size from a finite population of size is chosen in such a way that each of the possible samples has the same probability, , of being selected.6.3 Statistics 统计量Definition 6.3.1 A statistic is a function of the observations in a sample which does not depend on any unknown parameters. We will discuss the definition of the word statistic and some commonly used statistics in this section. Sample mean子样均值Sample variance子样方差Definition 6.3.2 Given is a random sample of size from a random variable , then(i) The sample mean is , (6.3.1) (ii) The sample variance is (6.3.2)and the sample standard deviation is . (6.2.3)Remark: The sample variance is sometimes defined as in some books. Here, and are both random variables. Suppose a set of values of observations of the random sample is , then the observation value of and are denoted as 观察值.In the future, normally we use capital letters to represent random variables and use small letters to represent the observation values.Theorem 6.2.1 If is the sample variance that is defined in (6.3.2), we can get that.顺序统计量Definition 6.2.3 Ranking the elements of the random sample in an increasing order to yield , where is the smallest, is the second smallest and so on. is called the minimum sample value and is the maximum sample value. is referred as the -th order statistic, With the definition of the order statistics, we are able to introduce some more terms that are also useful and important. 中位数 Definition 6.2.4 If a random sample has the order statistics , then(i) The Sample Median is (ii) The Sample Range is .Example 6.2.1 is a random sample of size 5. If an observation of this sample yields the values 2, 5, 1, 4, 8. Then, we can get the value of statistics as the following. Sample mean: Sample variance: : 1, 2, 4, 5, 8Sample median: Sample range: 6.4 Sample Distributions 抽样分布It should be kept in mind that a statistic, being computed from samples, is a random variable. 1sampling distribution of the mean 均值的抽样分布Theorem 6.3.1 If is mean of the random sample of size from a random variable which has mean and the variance , then and .Proof. First,. Second, since each pair of and , with , are independent. We can get that . It is customary to write as and as . Here, is called the expectation of the mean.均值的期望 is called the standard error of the mean. 均值的标准差This formula shows that the standard deviation of the distribution of decreases when , the sample size, is increased. It means that when becomes larger, we can expect that the value of to be closer to .方差越小表示子样均值越靠近总体均值Corollary 6.4.1 If is the mean of a random sample of size from a population that is a normal distribution , then .总体服从正态分布X ，则 ，总体分布未知， 但n 够大，则 N(, 2/n) （子样均值也近似服从正态）Example 8-1find P68.9Solution. Since , ， so P68.9= =PZ 2.8 =1PZ 2.8=0.5 0.4974=0.0026Definition 6.4.1 The mean and the variance of the finite population are .Theorem 6.4.3 If is mean of a random sample of size from a finite population of size with the mean and the variance , then and .Distribution of the sample standard Deviation（子样标准差的分布） the population with mean=m standard deviation= the sample standard deviation for large n , 此处并没有给出具体的分布，但告诉我们，当n足够大时，*子样标准差的均值 ,和子样标准差的标准差 Homework 6.1 6.6 6.8 堂上练习设总体服从正态分布N(12,4) ,今抽取容量为5的样本X1,X2,X3,X4,X5，试问：（1） 样本均值 大于13的概率是多少？（2） 样本均值的数学期望E、方差D、子样方差的数学期望是多少？（3） 如果（1，0，3，1，2）是样本的一个观察值，它的样本平均值和样本方差等于多少？数据处理（找数据的特征、规律）1直方图 histogramExample 6.3.2 We are interested in the distribution of peoples age in some city. In our hand, we have a sample of the ages of 50 people who were randomly picked and the data are listed below:3523482115361231454 21439215325715562246478501194328253415223051881675222741334561352843593890706Find the frequency table and draw the histogram.Solution. We find that the minimum sample value in this example is and the maximum sample value is . So, we choose the range of the frequency table as and divided it into 10 classes: 0, 10), 10, 20), 20, 30), 30, 40), 40, 50), 50, 60), 60, 70), 70, 80), 80, 90), 90, 100.Then, we count the number of sample values in each class and get the following frequency table.AgeFrequency0, 10)410, 20)720, 30)1030, 40)840, 50)650, 60)560, 70)370, 80)480, 90)190, 1002Table 6.3.1At last, using the horizontal axis to represent the classes and the vertical axis to represent the frequency, we can draw the histogram below. 214356810740 10 20 30 40 50 60 70 80 90 100Figure 6.3.1 In summery, for a random sample of size , the process of drawing a histogram is as following: 方法步骤(1) Find the minimum sample value and maximum sample value . Decide the number of classes (usually between 5 and 20) and then decide a fitting range of the frequency table so that is a little less than and is a little bigger than . (2) Insert points into : so that the range is divided into equal length classes: , , , . The length of each class is .(3) Count for each class how many sample values are inside this class, and then get the frequency table.(4) Using the horizontal axis to represent the classes and the vertical axis to represent the frequency, draw the histogram at last.2带频数的样本均值和样本方差For Table 6.3.1 , How to find the sample mean and sample variance? Table 6.3.1AgeFrequency0, 10)410, 20)720, 30)1030, 40)840, 50)650, 60)560, 70)370, 80)480, 90)190, 1002Definition 6.3.5 In a frequency table with classes, if the class midpoints are and the respective frequencies are , then(i) The sample mean is defined as (6.3.6)(ii) The sample variance is (6.3.7)(iii) The standard deviation is (6.3.8)where .Example 6.3.2 Find the sample mean and the sample variance if we have a random sample that is given in Table 6.3.1.Solution. By Definition 6.3.5, we get that*6.5 Chi-square Distributions 1gamma distribution, Definition 6.5.1. A random variable has a gamma distribution if its probability density is given by (6.5.3)where .gamma function for .Gamma function has a few useful properties, such the recursion formula.Also, when is a positive integer, we have.At last, an important special value is Properties 性质Theorem 6.5.1 The mean and the variance of the gamma distribution are given by and .Theorem 6.5.2 If are independent random variables and , then.（有可加性）The gamma densities with several special values of and are shown in Figure 6.5.1. The readers can get some idea about the shape of the gamma distribution.Figure 6.4.1: Graphs of gamma distribution 2特例（1） and ，- exponential distributionthe probability density of is （2） and ，- chi-square distribution Definition 6.5.2 A random variable has a chi-square distribution if its probability density is given by . (6.5.4)开方分布 Chi-square distribution 卡方分布the parameter - degree of freedom（自由度）. We will write if is a random variable which follows a chi-square distribution with the degree of freedom .Corollary 6.5.1 The mean and the variance of the chi-square distribution is given by and . The chi-square distribution is closely related to the normal distribution and has many important applications in statistics. Let us list several meaningful properties below.卡方分布与正态分布有密切联系，在统计学中有重要的应用Theorem 6.5.3 Let has the standard normal distribution . Then follows the chi-square distribution with degree of freedom , in short, .Theorem 6.5.4 If are independent random variables that each of them has standard normal distribution , then.Theorem 6.5.5 If are independent random variables and for , then.Theorem 6.5.6 Suppose and are two independent random variables. If and with . Then . 3重要结论Theorem 6.4.7 Suppose and are the sample mean and the sample variance of a random sample of size from a population that follows a normal distribution . Then(i) and are independent, and(ii) .Proof. We omit the proof of part (i) since it is beyond the scope of this book. Let us show only the part (ii). In order to study the distribution of , we need the identity (6.5.5)In fact, the left hand side of (6.4.5) isand the right hand side of (6.5.5) is which proves (6.4.5). By the definition of in (6.3.2), we have . Substitute it into (6.4.5) and then divide both sides of (6.5.5) by to get (6.5.6) We know that and from Theorem 6.5.4, we conclude that . At the same time, by Theorem 6.4.1 and Theorem 6.5.3, we get that . Therefore, thanks Theorem 6.4.6, we have that . Let be a random variable whose distribution is . Find the value such that 6.6 Students Distributions (t-Distribution)学生分布/*Consider the case that a random sample of size from a normal population with the mean and the variance . In Corollary 6.3.1, we know that the random variable is also a normal distribution . Furthermore, . (6.5.1)Of course this is an important conclusion in statistics. However, in application, the population standard deviation is usually unknown. Therefore, people start to seek a replacement of with an estimate. Naturally, the sample standard deviation seems to be a good choice. Since in the later material, we will know that , we replace in (6.5.1) by and it comes the problem: What is the exact distribution of for a random sample from a normal population? This problem was originally studied by W.S. Gosset, who wrote under the pen name “student” because the company where he worked, Guiness Brewery in Dublin, did not allow publication by the employees. Gosset found that the quantities resulting from this substitution no longer follows normal distribution, instead, it satisfied a different type of distribution which has been called Students t-distribution since then. Let us first introduce a more general situation which leads to the formal definition of t-distribution. */1Students distributionTheroem 6.6.1 Let be a standard normal random variable and be a random variable with degree of freedom such that and are independent, then the random variable has density function for and it is called the t-distribution with degrees of freedom. If , then We omit the detailed proof which is above the requirement of this book. If a random distribution has the t-distribution with degrees of freedom, we will write for short.Figure 6.6.1 Figure 6.6.1 includes the graphs of standard normal distribution, t-distributions with 1 and 5 degrees of freedom. We can see that the curves of t-distributions resemble in general shape the normal distribution. Also, as increases, the t-distribution will get closer to the normal distribution. In fact, the standard normal distribution can be considered to be the limiting case for the distributions, that is, .With the help of Theorem 6.6.1, we are now able to obtain the distribution of which is the problem raised in the beginning of this section. It is treated in the following theorem.2. 应用Theorem 6.6.2 Suppose that and are the mean and the variance of a random sample of size from a population which is a normal distribution . Then has the t-distribution with degrees of freedom. In short,.Proof. Let and .By the Corollary 6.4.1 and Theorem 6.5.7, we get that and , also, and are independent. Thusand by Theorem 6.5.1, we conclude that . Let be a -distribution with degrees of freedom , find the value such that(a) , when ;(b) , when .Homework 6.9, 6.14, 6.176.6 F-DistributionsIn this section, we will study another important distribution in practical applications of statistics, the F-distribution. It helps us to deal with the comparison of the variability of two samples. In order to obtain the F-distribution, we consider the ratio of two independent chi-square random variables, each divided by its own degrees of freedom. The distribution of this radio is presented in the following theorem.Theorem 6.6.1 Let and be two independent random variables such that and . Then the random variable has the density function